which are also orthogonal

(0.003 seconds)

21—28 of 28 matching pages

21: 28.32 Mathematical Applications

… ►Also let

\mathcal{L}

be a curve (possibly improper) such that the quantity … ►Kernels

K

can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. … ►is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which

A, B

are separation constants. …

22: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►For these and other results, and also cases in which any one of

j_{1},j_{2},j_{3}

\frac{3}{2}

2

, see Edmonds (1974, pp. 125–127). … ►See also Micu (1968), Louck (1958), Schulten and Gordon (1975a), Srinivasa Rao and Rajeswari (1993, pp. 220–225), and Luscombe and Luban (1998). ►

§34.3(iv) Orthogonality

… ►Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to

\mathit{3j}

symbols, for which see Edmonds (1974, Chapter 4). …

23: 3.8 Nonlinear Equations

… ►We continue with

x_{2}

and either

x_{0}

x_{1}

, depending which of

f_{0}

and

f_{1}

is of opposite sign to

f(x_{2})

, and so on. … ►For the computation of zeros of orthogonal polynomials as eigenvalues of finite tridiagonal matrices (§3.5(vi)), see Gil et al. (2007a, pp. 205–207). … ►However, to guard against the accumulation of rounding errors, a final iteration for each zero should also be performed on the original polynomial

p(z)

. … ►Newton’s rule can also be used for complex zeros of

p(z)

. … ►These results are also useful in ensuring that no zeros are overlooked when the complex plane is being searched. …

24: 31.11 Expansions in Series of Hypergeometric Functions

… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►

\lambda

\mu

must also satisfy the condition … ►For example, consider the Heun function which is analytic at

z=a

and has exponent

\alpha

\infty

. … ►

\mu=\gamma+\delta-2.

…

25: 14.30 Spherical and Spheroidal Harmonics

… ►Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics

R_{n}^{m}(x)=e^{-i\pi n/2}P^{m}_{n}\left(ix\right)

and

T_{n}^{m}(x)=ie^{i\pi n/2}Q^{m}_{n}\left(ix\right)

which are real when

x>0

and

n=0,1,2,\dots

. … ►

Orthogonality

… ►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►See also (18.18.9) and (34.3.19). … ►has solutions

W(\rho,\theta,\phi)=\rho^{l}Y_{{l},{m}}\left(\theta,\phi\right)

, which are everywhere one-valued and continuous. …

26: 18.35 Pollaczek Polynomials

… ►

§18.35(ii) Orthogonality

… ► … ►Then … ►For type 3 orthogonality (18.35.5) generalizes to … ►These polynomials also occur in connection with the Coulomb problem, see §18.39(iv).

27: 3.11 Approximation Techniques

… ►Also, let … ►See also Cody (1970) and Ralston (1965). … ►They enjoy an orthogonal property with respect to integrals: … ► … ►In computer graphics a special type of spline is used which produces a Bézier curve. …

28: 3.5 Quadrature

… ►See also Poisson’s summation formula (§1.8(iv)). … ►which depends on function values computed previously. … ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ► … ►See also Gil et al. (2003b). …